erdős #861

← #860 · #862 → (packet.json; erdosproblems.com)

Let $f (N)$ be the size of the largest Sidon subset of ${1, \dots, N}$ and $A (N)$ be the number of Sidon subsets of ${1, \dots, N}$ . Is it true that $A (N) / 2^{f (N)} \to \infty ?$ Is it true that $A (N) = 2^{(1 + o (1)) f (N)} ?$

Worked, still open.

number theory · solved · 0 attempts

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A003022 — Length of shortest (or optimal) Golomb ruler with n marks.1,3,6,11,17,25,34,44,55,72,85,106,127,151,177,199,216,246,283,333,356,372,425,480,492,553,585 A143823 — Number of subsets {x(1),x(2),...,x(k)} of {1,2,...,n} such that all differences |x(i)-x(j)| are distinct.1,2,4,7,13,22,36,57,91,140,216,317,463,668,962,1359,1919,2666,3694,5035,6845,9188,12366,16417,21787,28708,37722,49083,63 A143824 — Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.0,1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10 A227590 — a(n) = A003022(n)+1 with a(1)=1.1,2,4,7,12,18,26,35,45,56,73,86,107,128,152,178,200,217,247,284,334,357,373,426,481,493,554,586

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Vela Sidon frontier (A309370) · verified work

A B₂/Sidon problem — the same object family as Vela's verified Sidon records, where nine improved terms were accepted into OEIS A309370 (the campaign's first external adoption).

OEIS A309370 ↗ · verified-combinatorics (witnesses + verify.py) ↗ · the Erdős campaign

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#14Let

A \subseteq N

. Let

B \subseteq N

be the set of integers which are representable in exactly one way as the sum of two elements from

A

.Is it true that for all

ϵ > 0

and large

N

∣{1, \dots, N} \ B ∣ ≫_{ϵ} N^{1/2 - ϵ} ?

Is it possible that

∣{1, \dots, N} \ B ∣ = o (N^{1/2})?

A143824 #30Let

h (N)

be the maximum size of a Sidon set in

{1, \dots, N}

. Is it true that, for every

ϵ > 0

h (N) = N^{1/2} + O_{ϵ} (N^{ϵ})?

A003022 #43If

A, B \subset {1, \dots, N}

are two Sidon sets such that

(A - A) \cap (B - B) = {0}

then is it true that

(2 ∣ A ∣) + (2 ∣ B ∣) \leq (2 f ( N )) + O (1),

where

f (N)

is the maximum possible size of a Sidon set in

{1, \dots, N}

? If

∣ A ∣ = ∣ B ∣

then can this bound be improved to

(2 ∣ A ∣) + (2 ∣ B ∣) \leq (1 - c + o (1)) (2 f ( N ))

for some constant

c > 0

?A003022 #155Let

F (N)

be the size of the largest Sidon subset of

{1, \dots, N}

. Is it true that for every

k \geq 1

we have

F (N + k) \leq F (N) + 1

for all sufficiently large

N

?A003022 #530Let

ℓ (N)

be maximal such that in any finite set

A \subset R

of size

N

there exists a Sidon subset

S

of size

ℓ (N)

(i.e. the only solutions to

a + b = c + d

S

are the trivial ones). Determine the order of

ℓ (N)

.In particular, is it true that

ℓ (N) \sim N^{1/2}

?A143824

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solved

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